Optimal. Leaf size=15 \[ -\frac {1}{a c (1+a x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6275, 32}
\begin {gather*} -\frac {1}{a c (a x+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 6275
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\frac {\int \frac {1}{(1+a x)^2} \, dx}{c}\\ &=-\frac {1}{a c (1+a x)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 3 vs. order 1 in
optimal.
time = 0.02, size = 18, normalized size = 1.20 \begin {gather*} -\frac {e^{-2 \tanh ^{-1}(a x)}}{2 a c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 16, normalized size = 1.07
method | result | size |
gosper | \(-\frac {1}{a c \left (a x +1\right )}\) | \(16\) |
default | \(-\frac {1}{a c \left (a x +1\right )}\) | \(16\) |
risch | \(-\frac {1}{a c \left (a x +1\right )}\) | \(16\) |
norman | \(\frac {\frac {x}{c}+\frac {a \,x^{2}}{c}}{\left (a x +1\right )^{2}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 14, normalized size = 0.93 \begin {gather*} -\frac {1}{a^{2} c x + a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 14, normalized size = 0.93 \begin {gather*} -\frac {1}{a^{2} c x + a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 12, normalized size = 0.80 \begin {gather*} - \frac {1}{a^{2} c x + a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (a x + 1\right )} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 13, normalized size = 0.87 \begin {gather*} -\frac {1}{a\,\left (c+a\,c\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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